Resolving the Missing Deflation Puzzle

We propose a resolution of the missing deflation puzzle. Our resolution stresses the importance of nonlinearities in price- and wage-setting when the economy is exposed to large shocks. We show that a nonlinear macroeconomic model with real rigidities resolves the missing deflation puzzle, while a linearized version of the same underlying nonlinear model fails to do so. In addition, our nonlinear model reproduces the skewness of inflation and other macroeconomic variables observed in post-war U.S. data. All told, our results caution against the common practice of using linearized models to study inflation and output dynamics.


Introduction
The Great Recession has left macroeconomists with many puzzles. One such puzzle is the alleged breakdown of the relationship between in ‡ation and the output gap -also known as the Phillips curve. The Great Recession generated an extraordinary decline in U.S. GDP of about 10 percent relative to its pre-crisis trend while in ‡ation dropped only by about 1.5 percent (see. e.g. Christiano, Eichenbaum and Trabandt, 2015). The modest decline in in ‡ation was surprising to many macroeconomists. For instance, New York Fed President John C. Williams (2010, p. 8) wrote: "The surprise [about in ‡ation] is that it's fallen so little, given the depth and duration of the recent downturn. Based on the experience of past severe recessions, I would have expected in ‡ation to fall by twice as much as it has".
The small drop in in ‡ation has been referred to as the "missing de ‡ation puzzle" and has attracted considerable interest among academics and policymakers. Hall (2011) argues that since in ‡ation fell so little in the face of the large contraction in GDP, one might view in ‡ation as being essentially exogenous to the economy. Speci…cally, Hall (2011) argues that popular DSGE models based on the simple New Keynesian Phillips curve "cannot explain the stabilization of in ‡ation at positive rates in the presence of long-lasting slack". Similarly, Ball and Mazumder (2011) argue that Phillips curves estimated for the post-war pre-crisis period in the United States cannot explain the modest fall in in ‡ation during the years of the …nancial crisis from 2008 through 2010. They argue that the …t of the standard Phillips curve deteriorates sharply during the crisis. A further challenge to the New Keynesian Phillips curve is raised by King and Watson (2012), who …nd a large discrepancy between actual in ‡ation and in ‡ation predicted by the workhorse Smets and Wouters (2007) model. Interestingly, all above contributions use linearized variants of the New Keynesian Phillips curve in their analysis.
Our proposed resolution of the missing de ‡ation puzzle rests on two key elements. First, we argue that it is key to introduce real rigidities in price-and wage-setting. To do this, we follow Dotsey and King (2005) and Smets and Wouters (2007) and use the Kimball (1995) aggregator instead of the standard Dixit-Stiglitz (1977) aggregator. The Kimball aggregator introduces additional strategic complementarities in the price-and wage-setting, which lowers the sensitivity of prices and wages to the relevant wedges for a given degree of price-and wage-stickiness. As such, the Kimball aggregator is commonly used in New Keynesian models, see e.g. Smets and Wouters (2007), as it allows to simultaneously account for the macroeconomic evidence of a low Phillips curve slope and the microeconomic evidence of frequent price changes.
Second, we argue that the standard procedure of linearizing all equilibrium equations around the steady state, except for the zero lower bound (ZLB) constraint on the nominal interest rate, introduces large approximation errors when large shocks hit the economy as was the case during the Great Recession. Implicit in the linearization procedure is the assumption that the linearized solution is accurate even far away from the steady state. Our analysis shows that the linearized solution is very inaccurate far away from the steady state. We show that one ought to use the nonlinear model solution instead of the linearized model solution to understand the output-in ‡ation dynamics during the Great Recession. We show that the nonlinearity implied by the Kimball aggregator is a key model feature that accounts for the di¤erences between the linearized and nonlinear model solutions: it is crucial to resolve the missing de ‡ation puzzle. The Kimball aggregator implies that the demand elasticity for intermediate goods is state-dependent, i.e. …rms'demand elasticity is an increasing function of its relative price and the demand curve is quasi-kinked. 1 While the fully nonlinear model takes the state-dependence of the quasi-kinked demand curve explicitly into account, a linear approximation replaces this key nonlinearity by a linear function. When the economy is exposed to large shocks the state-dependence of the quasi-kinked demand curve becomes quantitatively important and the linear approximation ceases to provide accurate results.
A key contribution of our work is that we provide a structural general equilibrium model which can account for the alleged breakdown of the Phillips curve during the Great Recession. 2 That is, a nonlinear formulation of our model generates a very modest fall of in ‡ation in the wake of a large and persistent contraction of output. By contrast, the linearized version of the same underlying nonlinear model predicts a much larger fall of in ‡ation and thereby fails to account for the outputin ‡ation dynamics in the data. Therefore, our model provides a resolution to the missing de ‡ation puzzle: the puzzle arises only if one uses the linearized solution to predict in ‡ation in the Great Recession. Section 2.1.2 provides the intuition and Figure 4 illustrates this key result graphically. In a nutshell, our model implies nonlinear price and wage Phillips curves, i.e. nonlinear relationships between price and wage in ‡ation and the output gap. The slopes of our price and wage Phillips curves are notably ‡atter in recessions than in booms, i.e. our Phillips curves have a banana-or boomerang-shape as in the seminal paper by Phillips (1958). Consequently and consistent with the data, our nonlinear model implies that in ‡ation falls relatively little in deep recessions. By contrast, the linearized model erroneously predicts a much larger fall in in ‡ation by extrapolating local decision rules around the steady state to deep recessions far away from the steady state. 3 We establish our main results in a variant of the Erceg, Henderson and Levin (2000) (EHL henceforth) benchmark model. The EHL model, however, does not allow for endogenous capital accumulation and other real rigidities like habit formation in consumer preferences and investment adjustment costs. Therefore, we examine the robustness of our results in an estimated mediumsized New Keynesian model based on Christiano, Eichenbaum and Evans (2005) and Smets and Wouters (2007) which includes the features listed above. We estimate this extended model using Bayesian maximum likelihood techniques. We show that the insights from the EHL model carry over to the estimated medium-sized model. In addition, by performing stochastic simulations with the estimated nonlinear model, we also establish two further important insights.
First, a large body of recent work -see for instance IMF (2016IMF ( , 2017 -has been aimed towards understanding the absence of upward pressure of price and wage in ‡ation during the recovery from the Great Recession, also called the missing in ‡ation puzzle. The nonlinear formulation of our model o¤ers an explanation for this phenomenon. Given the nonlinear Phillips curves in our model, price and wage in ‡ation remain subdued until the level of economic activity has recovered su¢ ciently relative to its potential. Put di¤erently, even though economic growth may resume after a deep recession, price and wage in ‡ation will only increase modestly until economic slack has subsided su¢ ciently. Therefore, our model does not only resolve the missing de ‡ation puzzle, it also o¤ers an explanation for the missing in ‡ation puzzle. According to our model, subdued price and wage in ‡ationary pressures as observed during the recovery from the Great Recession are indicative of substantial economic slack. 4 While the linearized model suggests that price and wage in ‡ation should rise sharply during a recovery from a deep recession, the nonlinear model suggests otherwise: price and wage in ‡ation will not rise until economic slack has narrowed su¢ ciently from its elevated recession level.
Second, our nonlinear model can be used to understand the positive skewness in post-war U.S. price in ‡ation, i.e. that price in ‡ation scares are much more common than de ‡ationary episodes. We establish this result by comparing the skewness for in ‡ation in the data and our model. Consistent with the data, our nonlinear model implies positive skewness in in ‡ation. In addition, our nonlinear model also generates positive skewness for wage in ‡ation as well as for the nominal policy interest rate as observed in the data. By contrast, the linearized version of the model fails to account for the skewness of these time series.
The paper is organized as follows. Section 2 presents the stylized New Keynesian model with stickiness and real rigidities in price-and wage-setting. Section 3 discusses the results based on the stylized model. Section 4 examines the robustness of our results in an estimated medium-sized New Keynesian model. Section 5 discusses related literature on micro-and macroeconomic evidence.
Section 6 provides concluding remarks.

A Stylized New Keynesian Model
The simple model we use is very similar to the EHL model with gradual price and wage adjustment.
We deviate from EHL in two ways. First, by allowing for Kimball (1995) aggregators in price-and wage-setting (with the standard Dixit and Stiglitz (1977)  Levin, Lopez-Salido and Yun (2007), we assume that G Y ( ) is given by the following concave and increasing function: is the standard Dixit and Stiglitz (1977) aggregator. If p < 0; G Y ( ) is the Kimball (1995) aggregator as used in e.g. Smets and Wouters (2007). 6 Note that G Y (1) = 1, implying constant returns to scale when all intermediate …rms produce the same amount. 5 The complete speci…cation of the nonlinear and linearized formulation of the model is provided in the Technical Appendix available at: https://sites.google.com/site/mathiastrabandt/home/downloads/LindeTrabandt_In ‡ation_TechApp.pdf 6 The parameter used in Smets and Wouters (2007) to characterize the curvature of the Kimball aggregator can be mapped to our model using the following formula: p = p p 1 p : Final goods producers minimize the cost of producing a given quantity of output Y t taking as given the price P f;t of each intermediate good Y f;t . Final goods producers sell units of …nal output at a price P t which they also take as given.
Formally, producers solve: (1). The …rst order conditions can be written as: (3) where " p = p (1+ p ) 1 p and # p t denotes the Lagrange multiplier on the aggregator constraint (1). Equation (2) denotes the demand for goods, equation (3) is the aggregate price index and equation (4) is the zero pro…t condition for …nal goods …rms. Note that for p = 0 we obtain the standard  Calvo-Yun (1996) style staggered nominal contracts. In each period, each …rm f faces a probability 1 p of being able to choose its pro…t maximizing price. Firms solve: where t+s is household marginal utility, & t+s is a household discount factor shock discussed in section 2.1.3 and demand Y f;t+s is given by equation (2).
If a …rm does not optimize its price in a given period, it updates its price according toP t = P t 1 , where is the steady-state gross in ‡ation rate, i.e. = 1 + where the is net steady state price in ‡ation rate.

Intuition
To provide intuition about the implications of the Kimball vs. Dixit-Stiglitz aggregator, we consider the ‡exible price version of the price setting problem of …rms.
The upper left panel in Figure 1 shows how relative demand in equation (2) is a¤ected by the relative price under alternative assumptions about P ; given a gross markup of p = 1:1 and assuming that # p t = 1. When p = 0, the demand curve exhibits the standard Dixit-Stiglitz constant demand elasticity, implying a log-linear relationship between relative demand and relative price, i.e. log y f = " p log p f where y f Y f =Y and p f P f =P: With the Kimball speci…cation p < 0 -as in e.g. Smets and Wouters (2007) -a …rm instead faces a quasi-kinked demand curve, implying that a drop in its relative price triggers only a small increase in demand. On the other hand, a rise in its relative price generates a larger fall in demand compared to the p = 0 case.
Relative to the standard Dixit-Stiglitz aggregator, this introduces more strategic complementarities (or real rigidities) in price setting.
The upper right panel of Figure 1  More strategic complementarities in the Kimball speci…cation imply that the pro…t function f (p f ) becomes notably more curved when p < 0, especially when the …rm's relative price exceeds unity.
The two middle panels in Figure 1 show how 5 percent positive and negative changes in marginal cost a¤ect …rm pro…ts f (p f ) and the optimal pro…t maximizing relative price p opt f : The left panel shows the e¤ects for the Kimball speci…cation and the right panel shows the e¤ects for the Dixit-Stiglitz speci…cation. The pro…t function shifts up when marginal cost decline and shifts down when marginal cost rise.
In the Dixit-Stiglitz case ( p = 0), the optimal relative price is changed by exactly the same percent change in marginal costs. This result re ‡ects the optimal markup pricing rule implied by maximizing pro…ts subject to the demand curve: Dixit-Stiglitz (nonlinear): p opt f = p mc: Log-linearizing around steady state yields: Dixit-Stiglitz (log-linear): where 'bar' variables denote steady state variables. Note that the nonlinear and log-linearized optimal pricing rules under Dixit-Stiglitz have the same implications: the optimal relative price changes one to one -in percent terms -with marginal cost.
With the Kimball speci…cation ( p < 0), the results can di¤er sharply from the Dixit-Stiglitz case. Maximizing pro…ts subject to the demand curve yields: Kimball (nonlinear): p opt where p denotes the net price markup, i.e. p = 1 + p : The left middle panel in Figure 1 shows that a negative shock to mc implies a smaller change in p opt f in absolute value (decline from 1 to 0.997) than a positive shock to mc (increase from 1 to 1.0045) in the nonlinear framework.
The asymmetric response of p opt f to negative and positive shocks to mc in the nonlinear model is apparent from equation (6). The equation implies that the asymmetry will be stronger the smaller (i.e. negative) p and the larger the change in mc.
What is the intuition for the asymmetric response of p opt f when p < 0? The nonlinearity embedded in the Kimball aggregator and the resulting nonlinear demand curve are key to understand the asymmetric price response. Consider a fall in marginal cost, i.e. a recession. Recall the pro…t function f (p f ) = (p f mc) y f (p f ): Keep in mind that the demand elasticity falls when the relative price of a …rm falls. Thus, at the margin, a …rm's ability to increase its demand y f (p f ) by cutting its price is limited, especially the deeper the recession. Large price cuts result in lower pro…ts because at the margin, demand y f (p f ) rises only by little while revenues p f y f (p f ) fall substantially. Therefore, …rms have little incentive to cut prices a lot. The incentive to cut prices becomes weaker the larger the fall in marginal cost since the lower relative price of a …rm reduces the demand elasticity implying less demand can be crowded-in by the …rm when cutting its price.
All told, …rms cut their price by less at the margin the deeper the recession, i.e. an asymmetric price response. By continuity, …rms change their prices by more at the margin the larger the boom in the economy.

mc mc mc
Note that in contrast to Dixit-Stiglitz, the optimal relative price does not move one to one in percent terms with marginal costs under the Kimball speci…cation. With Kimball, …rm's adjust their prices less than with Dixit-Stiglitz since

Households and Wage Setting
Labor Contractors. Competitive labor contractors aggregate specialized labor inputs N j;t supplied by households into homogenous labor N t which is hired by intermediate good producers. Labor contractors maximize pro…ts max t is the wage paid by the labor contractor to households for supplying type j labor. W t denotes the wage paid to the labor contractor for homogenous labor. Pro…t maximization is subject to is the Kimball aggregator speci…cation as used in Dotsey and King (2005) or Levin, Lopez-Salido and Yun (2007) adapted for the labor market. Note that ! w = (1+ w ) w 1+ w w where w 1 denotes the gross wage markup and w 0 is the Kimball parameter that controls the degree of complementarities in wage-setting.
Let # w t denote the multiplier on the labor contractor's constraint. Optimization results in the following optimality conditions: where " w = w (1+ w ) 1 w : Equation (7) denotes the demand for labor, equation (8) is the aggregate wage index and equation (9) is the zero pro…t condition for labor contractors. Note that for w = 0 we get the standard Dixit-Stiglitz demand curve and aggregate wage index.
Households. There is a continuum of households j 2 [0; 1] in the economy. Each household supplies a specialized type of labor service j to the labor market, for which it is a monopolistic supplier. The j th household maximizes where the choice variables of the j th household are consumption C j;t and risk-free government debt B t . The j th household also chooses the wage W j;t subject to Calvo sticky prices as in EHL. The household understands that when choosing W j;t it must supply the amount of labor N j;t demanded by a labor contractor according to equation (7).
The variable & t is an exogenous shock to the discount factor 0 < < 1. We assume that is exogenous with = 1 in steady state. P t denotes the aggregate price level. i t denotes the net nominal interest rate on bonds purchased in period t 1 which pay o¤ in period t: R k t is the rental rate of the …xed aggregate capital stock that the households rents to goods producing …rms.
T j;t are lump-sum taxes net of transfers and j;t denotes the share of pro…ts that the household receives. A j;t denotes payments and receipts associated with insurance in the presence of wage stickiness. 7 ! > 0 and 0 are parameters.
Utility maximization for consumption and government bond holdings yields the standard aggregate consumption Euler equation: Wage Setting. Households face a standard monopoly (labor union) problem of selecting W j;t to maximize utility (10) subject to the demand for labor (7). Following EHL, we assume that households experience Calvo-style frictions in its choice of W j;t : In particular, with probability 1 w the j th household has the opportunity to choose its optimal wage W opt j;t . With the complementary probability, the household must set its wage rate according toW j;t = w W j;t 1 where w denotes the steady state gross rate of wage in ‡ation. Households choose the optimal wage by maximizing subject to labor demand given by equation (7).

Monetary and Fiscal Policy
We assume that the central bank sets the nominal interest rate following a Taylor-type policy rule that is subject to the zero lower bound: where Y pot t denotes the level of output that would prevail if prices and wages were ‡exible.
In terms of …scal policy, we assume that the government balances its budget each period using lump-sum taxes.

Aggregate Resource Constraint
It is straightforward to show that the aggregate resource constraint is given by The variables p t 1 and w t 1 denote the Yun (1996) aggregate price and wage dispersion terms. Both price-and wagedispersion, ceteris paribus, will lower aggregate output in the economy. The Technical Appendix provides recursive formulations for the sticky price and wage distortion terms p t and w t . Note that p t and w t vanish when the model is linearized.

Parameterization
Time is taken to be quarters. We set the discount factor = 0:9975 and the steady state net in ‡ation rate = 0:005 implying a steady state nominal interest rate of i = 0:0075 (i.e., three percent at an annualized rate). We set the capital share parameter = 0:3 and the disutility of labor parameter = 0: As a compromise between the low estimate of the gross price markup p in Altig et al. (2011) and the higher estimated value by Smets and Wouters (2007), we set p = 1:1.
Consistent with microeconomic evidence in e.g. Nakamura and Steinsson (2008) we set p = 0:66; i.e. prices change on average every three quarters. To pin down the Kimball parameter p consider the log-linearized New Keynesian Phillips curve in our model: where d mc t denotes the log-deviation of real marginal cost from its steady state.^ t denotes the log-deviation of gross in ‡ation from its steady state. The parameter p denotes the slope of the Phillips curve and is given by The macroeconomic evidence suggest that the sensitivity of aggregate in ‡ation to variations in real marginal cost is very low, see e.g. Altig et al. (2011). To capture this, we set the Kimball parameter p = 12:2 so that the slope of the Phillips curve is p = 0:012 given the values for , p and p discussed above. 8 For the parameters pertaining to the nominal wage setting frictions we assume that w = 1:1, w = 0:75, and w = 6. These parameter values correspond roughly to those set and estimated in the medium-sized New Keynesian model discussed below. We use the standard Taylor (1993) rule parameters = 1:5 and x = 0:125.
To facilitate comparison between the nonlinear and linearized model, we specify an AR (1) process for the discount factor so that there is no loss in precision due to an approximation: where = 1. Our baseline parameterization adopts a persistence coe¢ cient = 0:95 capturing the persistent drop in output during the Great Recession.

Solving the Model
Our baseline results for the linearized and nonlinear model are based on the Fair and Taylor As a practical matter, we feed the equilibrium equations of the nonlinear and linearized model into Dynare. Dynare is a pre-processor and a collection of MATLAB routines which can solve nonlinear and linearized dynamic models with forward looking variables. The details about the implementation of the algorithm used can be found in Juillard (1996). We use the perfect foresight/deterministic simulation algorithm implemented in Dynare using the 'simul'command. 10 The algorithm can also easily handle the ZLB constraint: one just writes the Taylor rule including the max operator in the model equations, and the solution algorithm reliably calculates the model solution in fractions of a second. Thus, apart from obtaining intuition about the various mechanisms embedded in dynamic models, there is no need anymore to linearize dynamic models to solve, simulate and work with them.

In ‡ation and Output Dynamics in the Stylized Model
In this section, we report our main results for the linearized and nonlinear solution of the model outlined in the previous section. In section 3.1, we study the joint output-in ‡ation dynamics for large adverse demand shocks, and in section 3.2 we consider the e¤ects of both positive and negative shocks in long simulations of the model. 9 The introduction of wage stickiness and Kimball aggregation in the labor market in the present paper (in addition to price stickiness and Kimball aggregation in the goods market as in Lindé and Trabandt, 2018) should temper the e¤ect of shock uncertainty in the nonlinear model even further. To the extent that allowing for shock uncertainty impacts notably the linearized solution, the di¤erences between the linearized and nonlinear solutions we report in this paper are conservative: they would be even larger if we had allowed for shock uncertainty.
1 0 The solution algorithm implemented in Dynare's simul command is the method developed in Fair and Taylor (1983).

A Recession Scenario
We …rst study the e¤ects of a large adverse demand shock. Following the literature on …scal multipliers (e.g. Christiano, Eichenbaum and Rebelo, 2011), the particular shock we consider is a large positive shock to the discount factor t . Speci…cally, we assume that " ;1 = 0:01 in (13) so that t increases from 1 to 1.01 in the …rst period and then gradually reverts back to steady state. Figure 2 reports the linear and nonlinear solutions for a selected set of variables, assuming that the economy is in the deterministic steady state in period 0, and then the shock hits the economy in period 1. In the Technical Appendix we report results for an extended set of variables. The left column of Figure 2 shows results when the ZLB is, hypothetically, not assumed to be binding, whereas the right column shows the e¤ects when the ZLB binds. As is evident from the left column, the same-sized shock has a rather di¤erent impact on the economy depending on whether the model is linearized or solved in its original nonlinear form. For instance, we see that while output falls more in the nonlinear model, price in ‡ation falls notably less than in the linearized solution.
In the right column in Figure 2, we report the e¤ects of the same shock, but now assume that the central bank is constrained by the ZLB on the policy rate. Important insights about the di¤erences between the linearized and nonlinear solutions can be gained. First, although the drop in the potential real rate (not shown) is about the same in both models, the linearized model generates a much longer liquidity trap because in ‡ation and expected in ‡ation fall much more, which in turn causes the actual real interest rate (not shown) to rise much more initially. The larger initial rise in the actual real interest rate -and thus the rise in gap between actual and potential real interest rates -triggers a larger fall in the output gap. 11 Even so, and perhaps most important, we see that price in ‡ation falls substantially less in the nonlinear model compared to the linearized model. After running counterfactual experiments when we linearize parts of the model only, we …nd that the bulk of the di¤erences between the linearized and nonlinear solutions is driven by the linearization of the price-and wage-setting block of the model.
It is also instructive to compare the solutions with and without imposing the ZLB. Comparing the linearized solutions, we …nd that imposing the ZLB results in a notably larger fall in output (from -3.5 to almost -7 percent) and de ‡ation in prices and wages (not shown). For the nonlinear solution, we …nd that imposing the ZLB (albeit admittedly so with a shorter duration compared to the linearized solution) does not a¤ect the price and wage in ‡ation paths much -they are essentially una¤ected. The main impact of imposing the ZLB in the nonlinear model is -apart 1 1 Real GDP also falls more in the linearized model than in the nonlinear model because the discount factor shock does not impact potential real GDP. from the interest rate path -a somewhat deeper output contraction. According to United States congressional budget o¢ ce (CBO), the output gap fell by roughly 6 percent during the Great Recession but PCE price in ‡ation (4-quarter change) never fell below 1 percent. Our nonlinear solution is roughly consistent with these facts. By contrast, the linearized model is associated with a notably more depressed path for in ‡ation which is counterfactual compared to the data. 12 The Kimball aggregator is key for shrinking the sensitivity of in ‡ation to the large adverse shock in economic activity in the nonlinear model. To show this, we solve our model under the assumption that …nal good and labor services are aggregated with the standard Dixit-Stiglitz constant elasticity demand schedule ( p = w = 0). Because this adjustment changes the slopes of the linearized price and wage Phillips curves, we adjusted p and w so that the Phillips curve slopes of the Dixit-Stiglitz speci…cation are identical to the benchmark calibration with the Kimball aggregator. In practice, this requires increasing p and w to about 0.9, respectively. With the Dixit-Stiglitz aggregator in both wage-and price-setting, Figure A.4 in the Technical Appendix shows that price in ‡ation would fall even more in the nonlinear model compared to the linearized model. So, the strategic complementarities (or real rigidities) introduced by the Kimball aggregator are essential for a muted in ‡ation response.

Phillips Curves
To understand the unconditional di¤erences in the dynamics implied by the linearized and nonlinear solutions, we now undertake stochastic simulations of the model for shocks to the stochastic discount factor t . We solve and simulate the linearized and nonlinear model for a long sample of 50,000 periods contingent on exactly the same sequence of shocks f" ;t g 50;000 t=1 in equation (13). However, we use somewhat di¤erent standard deviations for the linearized model ( = 0.0014) and the nonlinear model ( = 0.0017), to ensure that the probability of hitting the ZLB is 10 percent in both model solutions. The left column in Figure 3 shows the paths of the …rst 10,000 periods with simulated data in the nonlinear model, whereas the right column shows the simulated data in the linearized model. In the Technical Appendix, we report results for an extended set of variables. rate. Subsequently, in ‡uential work by Roberts (1995), Clarida, Galí and Gertler (1999) and Blanchard (2000) has extended Phillips'approach to the relationship between price in ‡ation and the output gap. Thus, we use the simulated data in Figure 3 to produce bivariate scatter plots between price (and wage) in ‡ation on the y-axis and the negative of the output gap on the x-axis.
The negative output gap is the di¤erence between potential output and actual output. By using the negative of the output gap, we derive a downward-sloping relationship as in Phillips (1958). Moreover, the nonlinear model also features stronger responses of price and wage in ‡ation in booms than the linearized model, lending support for in ‡ation scares in booms (see e.g. Goodfriend, 1993). Another di¤erence between the linearized and nonlinear solutions is that the output gap is more volatile in the linearized solution, mainly due to strong propagation of the ZLB constraint.
Finally, it is imperative to understand that the relationships in Figure 4 are contingent on the assumption that the discount factor shock is the single driver of business cycles. No other shocks are assumed to a¤ect the economy. This is why we obtain a tight negative relationship between price and wage in ‡ation and economic slack. As we will see in the estimated model that we study next, this tight negative relationship ceases to exist in both the linearized and nonlinear model when di¤erent types of shocks a¤ect the economy simultaneously.

An Estimated Medium-Sized New Keynesian Model
The benchmark EHL model studied so far is useful to understand how nonlinearities in real rigidities in price-and wage-setting a¤ect in ‡ation dynamics in deep recessions. Speci…cally, we used the EHL model to demonstrate some of the bene…ts of taking nonlinearities into account as opposed to the traditional approach which entails log-linearizing the key model equations apart from the monetary policy rule.
However, this analysis was done in a stylized model with one shock and without allowing for e.g. endogenous capital accumulation. In this section we move on to a substantive analysis with the aim of examining the importance real rigidities in a nonlinear setting in a more quantitatively realistic model environment. We specify and estimate a medium-sized New Keynesian model with endogenous capital accumulation that follows closely the seminal model of Christiano, Eichenbaum and Evans (2005) but allows for variety of shocks as in Wouters (2003, 2007).
Next, we use the estimated model to examine the properties of the nonlinear and linearized solutions of it along several dimensions. First, the paths of model variables of the linearized and nonlinear models are compared with the actual outcomes during the Great Recession following Christiano, Eichenbaum and Trabandt (2015). Second, we redo the Phillips curve analysis in Section 3.2 using the estimated model. Third and …nally, we examine the ability of the estimated model to capture the unconditional skewness of price and wage in ‡ation as well as the skewness in other macroeconomic data.

Medium-Sized Model
Following Christiano, Eichenbaum and Evans (2005) and Wouters (2003, 2007), the medium-sized model includes both sticky nominal wages and prices. As in the EHL model, the Kimball (1995) aggregator is used to aggregate intermediate goods and labor to …nal output goods and e¤ective labor input. The medium-sized model also features internal habit persistence in consumption, and embeds a Q theory investment speci…cation where changing the level of investment is costly. 14 We use the same seven shocks as in Smets and Wouters (2007), i.e. shocks to mone-tary policy, government consumption, stationary neutral technology, stationary investment-speci…c technology, risk premium, price-and wage-markups. We relegate a more detailed description of the model to Appendix A.

Estimation
We now proceed to discuss how the model is estimated using U.S. data from 1965Q1-2007Q4. We estimate the linearized model with Bayesian maximum likelihood techniques. To solve and estimate the model we use Dynare. We estimate a similar set of parameters as Smets and Wouters (2007).

Data
We use seven key macroeconomic quarterly U.S. time series as observable variables: the log differences of real per capita GDP, consumption and investment; log-di¤erences of compensation per hour and the PCE de ‡ator; log deviations of hours per capita from its average as well as the Federal Funds rate. Further details about the data and the measurement equations linking the model variables to their data counterparts are provided in the Technical Appendix.

Estimation Methodology
Following Wouters (2003, 2007), we use full information Bayesian techniques to estimate the model. Bayesian inference starts out from a prior distribution that describes the available information prior to observing the data used in the estimation. The observed data is subsequently used to update the prior, via Bayes'theorem, to a posterior distribution of the model's parameters which can be summarized in the usual measures of location (e.g. mode or mean) and spread (e.g.

standard deviation and probability intervals). 15
Some of the parameters in the model are kept …xed throughout the estimation procedure (i.e., are subject to priors with a degenerate distribution). We choose to calibrate the parameters that are weakly identi…ed by the data that we use in the estimation. Table 1 provides the calibrated parameters which are set to standard values from the literature, whereas Table 2 contains steady states and implied parameters of the estimated model. Finally, Table 3 contains information about the parameter prior and posterior distributions.
implication, there is no intrinsic persistence in the linearized price and wage Phillips curves (i.e. they are completely forward-looking). 1 5 We refer the reader to Wouters (2003, 2007) for a more detailed description of the estimation procedure.

Role of Nonlinearities During the Great Recession
To illustrate the scope of nonlinearities for accounting for the missing de ‡ation puzzle, we use the estimated parameters and subject the nonlinear and linearized model to a positive risk premium shock. We compare the resulting model paths with the outcomes in the data following the methodology of Christiano, Eichenbaum and Trabandt (2015), CET henceforth. The risk premium shock RP;t enters the model in the optimality condition for bondholdings: where t denotes the Lagrange multiplier on the household budget constraint and R t is the gross nominal interest rate. RP;t denotes the risk premium shock used in Wouters (2003, 2007) which follows an estimated AR(1) process. Via its e¤ect on the Lagrange multiplier t , the risk premium shock also a¤ects the optimality conditions for investment and capital and enables the model to reproduce the strong positive correlation between consumption and investment observed in the data. Both CET and Lindé, Smets and Wouters (2016) argue that the risk premium shock was a key shock driving the Great Recession.
The risk premium RP;t in eq. (14) is assumed to rise in a uniform fashion for 16 quarters before gradually receding. Its size is set so that both the linearized and nonlinear models'output path roughly matches the "actual outcome" (discussed in detail below) during the crisis. Figure  To characterize what the data would have looked like absent the Great Recession, we follow CET and extrapolate a trend line for each variable for the period 2008Q3-2015Q2. Following CET we calculate the so-called "target gaps" as the di¤erence between the actual data and the …tted precrisis trends. The min-max ranges of the target gaps correspond to the gray intervals displayed in Figure 5 and the black solid line depicts the mean of the target gaps.
The purpose of our analysis is to assess whether the nonlinear and/or the linearized model is able to generate dynamics for the endogenous model variables that lie within the target gap ranges for the post 2008Q2-period. As can be seen from Figure 5, the elevated risk premium shock exerts a signi…cant adverse impact on the economy in which economic activity dampens and in ‡ation falls. As a result, the policy rate is driven towards a prolonged episode at the zero lower bound.
The model matches well the decline in consumption, but the fall of investment is somewhat smaller than in the data, presumably because the model lacks …nancial friction ampli…cation mechanisms.
Importantly, for the same-sized output response, price and wage in ‡ation in the nonlinear model fall by about one percentage point less than in the linearized model, con…rming our results in the benchmark EHL model.

Phillips Curves in the Estimated Model
To provide intuition for the muted in ‡ation response in the nonlinear model following a positive risk premium shock, Figure 6 shows a scatter plot of price in ‡ation and the negative output gap in both the linearized and the nonlinear variant of the estimated medium-sized New Keynesian model. Following the procedure in Section 3.2 to simulate data from the model, the upper left scatter plot is generated by sampling only risk premium innovations from a normal distribution using the estimated posterior mode and then simulating a long sample of 50,000 periods. Notice that the simulations are initiated at the steady state, and that the negative of the output gap is plotted on the x-axis, which means that a large positive number is associated with a deep recession.
As can be seen from the upper left plot in Figure 6, the linearized model is associated with a linear relationship between in ‡ation and the output gap when only risk premium shocks are used in the simulation, whereas the nonlinear model suggests a banana-or boomerang-shape relationship.
The relationship in the linearized model is entirely linear because the risk premium shocks are by themselves not large enough in the estimated model to drive the economy into a liquidity trap.
Even so, the risk premium shock is a key driver of business cycle dynamics in the estimated model and the di¤erence between the linearized and nonlinear solution has important implications for the dynamics of in ‡ation and output. Figure 6 also contains the implied price Phillips curves for the other six shocks of the model: we run stochastic simulations for each of the shocks -one at a time -in both the linearized and nonlinear model and then use the simulated data to construct the corresponding scatter plots. To do this, we use the estimated shock processes and parameters reported in Table 3. 16 In the bottom 1 6 Interestingly, equally-sized markup shocks have notably larger e¤ects on in ‡ation in the nonlinear model compared to the linearized model. We shrink the size of price and wage markup shocks in the nonlinear model such that the unconditional volatility of price and wage in ‡ation is the same in the nonlinear and linearized model. In other words, the nonlinear model does not depend as much on large markup shocks as the linearized model to explain the volatility of price and wage in ‡ation. In this sense, the nonlinear model is less prone to the critique against New Keynesian models by Chari, Kehoe and McGrattan (2010). right panel of Figure 6, we plot the results when all shocks are used in the model simulation.
As can been seen from Figure 6, risk premium and wage markup shocks account for the bulk of the volatility of in ‡ation and output in the estimated model. For these two shocks, we obtain the most sizeable ‡uctuations in in ‡ation and the output gap. Because these two shocks move the equilibrium far away from the steady state, the nonlinearities are also most evident for these two shocks. Given the estimated parameters, the wage markup shocks are the only source of ‡uctuations which have the potential to generate close to zero in ‡ation or very mild de ‡ationary episodes. It is striking how the dynamics of the price and wage markup shocks di¤er between the linearized and nonlinear solution. Other shocks which only cause moderate ‡uctuations in the output gap and in ‡ation result in small di¤erences between the linearized and nonlinear solutions because they do not generate any substantial deviations from the steady state.
Moreover, it is evident that there are no relevant trade-o¤s in stabilizing output and in ‡ation to ‡uctuations in risk-premium, government spending, and neutral technology shocks. By contrast, investment-speci…c and markup shocks -especially wage markup -create noticeable trade-o¤s between in ‡ation and output gap stabilization. Furthermore, when using all shocks in the simulation, the importance of the wage markup shock renders the Phillips curve completely ‡at or even upward-sloping, consistent with the empirical observation of no clear unconditional Phillips curve pattern in post-war U.S. data.
It is important to understand that many shocks do not generate any noticeable di¤erences between the linearized and nonlinear solutions when simulated one at a time since they do not drive the economy far away from the steady state. Note, however, that this does not imply that these shocks cannot propagate di¤erently in a recession (or boom) in the nonlinear model relative to the linearized solution. Figure 7 shows the results corresponding to those in Figure 6 conditional on a deep recession with an expected 8-quarter liquidity trap.
To construct Figure 7, we …rst simulate a baseline using an adverse risk premium shock which generates a recession with an anticipated 8-quarter liquidity trap. Relative to the baseline, we run 50.000 stochastic simulations of a counterfactual scenario in which other shocks are drawn from their estimated stochastic processes. We then compute the di¤erences between the 50.000 counterfactual scenarios and the baseline after one year. 17 Figure 7 shows notable di¤erences between the linearized and nonlinear solutions for many of the shocks for which Figure 6 reported no di¤erences, e.g. monetary policy, government spending, neutral technology and investment-speci…c technology shocks. According to the results in Figure   7, monetary policy is much less potent to a¤ect in ‡ation in a prolonged liquidity trap than in normal times. The e¤ects of monetary policy shocks on output are also somewhat moderated in the nonlinear model compared to the linearized solution. Negative investment-speci…c shocks, on the other hand, have notably more negative e¤ects on the output gap in a liquidity trap. In contrast to wage markup shocks, price markup shocks have essentially no e¤ect in a liquidity trap in the nonlinear model: …rms are unwilling to strongly respond to desired markup variations with Kimball aggregation.
It is important to point out that the results in Figure 7 generally imply notably ‡atter Phillips curves for many shocks in a liquidity trap. Hence, the nonlinear model o¤ers a possible explanation to the empirical observation that the sensitivity of in ‡ation to economic activity has become smaller in linearized models since the onset of the crisis (see e.g. Lindé, Smets and Wouters, 2016). In the nonlinear model, however, the smaller sensitivity is only temporary since it is contingent on a persistently negative output gap. The sensitivity will increase when economic slack has diminished su¢ ciently after the recession.

Accounting for Skewness in the Data
We have documented that the nonlinear model can account for the dynamics of in ‡ation during the Great Recession, including the ‡atter slope of the Phillips curve. We now turn to stochastic simulations to examine whether the nonlinear model may also help to explain unconditional moments of in ‡ation and other macroeconomic data. It is well known that in ‡ation has a positive skew during the postwar period, i.e. there are episodes with in ‡ation bursts and then there are episodes with very low and moderate rates of in ‡ation but no long-lived de ‡ationary episodes. This positive skew is a robust …nding for di¤erent measures of in ‡ation. The gray area in Figure 8  i.e. the observations plotted in the bottom right panel in Figure 6. As can be seen from Figure   8, the nonlinear model …ts the unconditional statistical properties of in ‡ation remarkably wellmuch better than the linearized model. The linearized model features a completely symmetric normal distribution for in ‡ation, with signi…cant mass in de ‡ationary territory. By contrast, the nonlinear solution features zero density in de ‡ationary territory for in ‡ation, in line with the data.
It is striking how much better the nonlinear model captures unconditional U.S. in ‡ation dynamics during the post-war period compared to the linearized model. 18 Figure 8 also depicts density plots for wage in ‡ation, real GDP growth and the Federal Funds rate. The nonlinear model captures the skewness of wage in ‡ation and the Federal Funds rate better than the linearized model. 19 The Technical Appendix also reports results for an extended set of variables.

Related Literature and Evidence
In this section, we discuss literature that is related to our work. We also discuss micro evidence that sheds light on the empirical relevance of real rigidities with a special emphasis on Kimball versus Dixit-Stiglitz aggregation. Finally, we also discuss macroeconomic evidence related to the missing de ‡ation puzzle.

Micro Evidence on Real Rigidities
A large body of literature has documented patterns of nominal price stickiness at a disaggregated good level, see e.g. Nakamura and Steinsson (2008), Klenow and Malin (2010) and the references therein. The empirical …ndings suggest that prices change on average about every three to four quarters, i.e. nominal stickiness is relatively short-lived. Christiano, Eichenbaum, and Evans (1999) and Gertler and Karadi (2015) have documented persistent real e¤ects of monetary policy shocks on output. Short-lived price stickiness and persistent real e¤ects of monetary shocks can be reconciled with each other in modeling frameworks with su¢ cient real rigidities, i.e. features that make …rms reluctant to change their prices by large amounts.
Amid this background, an empirical literature has started to emerge recently to examine whether there are quantitatively important real rigidities present in the data, see e.g. Beck and Lein (2015), Dossche, Heylen and Van den Poel (2010), Klenow and Willis (2016), and Levin and Yun (2008). 20 In general, the empirical literature appears to provide evidence in favor of Kimball aggregation rather than for Dixit-Stiglitz aggregation. That is, the empirical literature has estimated demand curves which exhibit state-dependent demand elasticities rather than constant demand elasticities.
Put di¤erently, there appears to be more support for quasi-kinked demand curves as implied by Kimball aggregation than for log-linear demand as implied by Dixit-Stiglitz aggregation (see the upper left panel in Figure 1 for an illustration).
However, although the evidence provided by the empirical literature supports Kimball over Dixit-Stiglitz aggregation in general, a balanced assessment of the evidence provided by the above papers is that it does not seemingly support the degree of real rigidities needed to …t the macro evidence. That said, the empirical literature has typically used micro data which does not cover the Great Recession period which is at the core of our study. In addition, real rigidities are di¢ cult to identify and di¢ cult to measure in the data as noted by Gopinath and Itskhoki (2010). Aggregate real rigidities imply a sluggish response of …rms' marginal costs to aggregate shocks. Firm-level real rigidities imply a muted response of …rms'prices conditional on price adjustment to marginal cost shocks. Since data on marginal costs is usually unavailable, it is hard to test these mechanisms directly. So, the evidence provided in e.g. Klenow and Willis (2016) is merely indicative, and their implied large idiosyncratic shock variance needed for their …ndings to be compatible with our estimate of the demand elasticity may be reconciled by allowing for …rm-speci…c customer relationships as in Levin and Yun (2008) and Kleshchelski and Vincent (2009).
All told, we interpret the micro evidence as providing at least partial support for our …ndings.
However, we believe that further empirical analysis based on joint micro data of goods prices, …rms'marginal costs and sales in a sample which also covers the Great Recession are needed before decisive conclusions can be drawn. We leave this analysis for future research.

Macro Evidence about the Missing De ‡ation Puzzle
Recent research has examined possible resolutions of the missing de ‡ation puzzle. Using standard time methods on U.S. data, an emering body of literature …nds support for the presence of nonlinearities in the Phillips curve. Doser et al. (2018) estimate a two-state regime Phillips curve and report a lower slope of the Phillips curve in a state in which the unemployment rate is elevated. Gagnon and Collins (2019) argue -based on time series evidence for a wide range of in ‡ation time series -that the Phillips curve is likely to be highly nonlinear when in ‡ation is low. The evidence provided in both papers is consistent with our structural model.
Another body of literature has used structural models to examine possible explanations for the missing de ‡ation puzzle. Lindé, Smets and Wouters (2016) and Fratto and Uhlig (2018) …nd that the Smets and Wouters (2007)  Our work is also related to Aruoba, Boccola and Schorfheide (2017). Using asymmetric price and wage adjustment costs, the authors show that their model can produce skewness in in ‡ation and output growth as observed in the data. However, their model cannot account for the skewness of the Federal Funds rate data while ours does. In contrast to these authors, our model does not rely on asymmetric adjustment costs but generates the skewness observed in macro data due to the Kimball (1995) aggregator. Moreover, Aruoba, Boccola and Schorfheide (2017) do not study the implications of the zero lower bound while our paper focuses on the interplay of large shocks and the zero lower bound to characterize nonlinearities in price and wage Phillips curves.
More generally, our work o¤ers an alternative and perhaps complementary explanation to understand the missing de ‡ation puzzle. Our resolution of the puzzle stresses the nonlinear in ‡uence of strategic complementarities and real rigidities in price-and wage-setting. We …nd it attractive due to its simplicity and due to its ability to address additional issues beyond the missing de ‡ation puzzle which we will discuss in the next section.

Conclusions
We have formulated a nonlinear macroeconomic model which goes a long way towards accounting for the missing de ‡ation puzzle, i.e. the empirical fact that in ‡ation fell little against the backdrop of a large and persistent fall in output during the Great Recession. Our resolution of the puzzle stresses the nonlinear in ‡uence of strategic complementarities and real rigidities in price-and wage-setting.
Our proposed nonlinear framework is also attractive along the following additional dimensions: (i) it mitigates the tension between the macroeconomic evidence of a low Phillips curve slope and the microeconomic evidence of frequent price changes; (ii) it allows to explain the empirical positive skew in in ‡ation without relying on a similar positive skew output (which is counterfactual); and (iii) it sheds light on the missing in ‡ation puzzle, i.e. why price and wage in ‡ation remained persistently low in the aftermath of the Great Recession. While our nonlinear model is capable of accounting for the above empirical facts, the linearized version of the underlying nonlinear model fails to do so. All told, our results caution against the common practice of using linearized models to study in ‡ation and output dynamics when the economy is exposed to large shocks.

Appendix A. The Estimated Medium-Sized New Keynesian Model
This appendix describes the estimated medium-sized model. The model closely follows Christiano, Eichenbaum and Evans (2005), Smets and Wouters (2007) and Christiano, Eichenbaum and Trabandt (2016).
Like the canonical sticky price and wage model described in Section 2, the medium-sized model includes monopolistic competition in the goods and labor markets and nominal rigidities in prices and wages with Kimball (1995) aggregation. In addition to saving in risk-free bonds, households can also save in physical capital. It takes one period before new investment turns into productive capital. In addition to the canonical model, the medium-sized model also features several real rigidities in the form of habit formation in consumption, investment adjustment costs, variable capital utilization, and …xed costs in production.
The model dynamics are driven by the same seven structural shocks as in Smets and Wouters (2007). The following four shocks a¤ect the actual economy but not the potential ( ‡exible price, ‡exible wage) economy: price-and wage markup shocks, risk premium shocks and monetary policy shocks. The following three shocks a¤ect both the actual and the potential economy: neutral technology shocks, investment-speci…c technology shocks and government spending shocks. All shocks follow AR(1) processes in logs, except for the markup shocks which follow ARMA(1,1) processes in logs and monetary policy shocks which are assumed to be white noise.
Below, we describe the households' and …rms' problems in the model, and state the market clearing conditions. The Technical Appendix which is available online provides a list of nonlinear and log-linearized equilibrium equations, measurement equations and further technical information.

A.1. Households
There is a continuum of households j 2 [0; 1] in the economy. Each household supplies a specialized type of labor j to the labor market. The j th household is the monopoly supplier of the j th type of labor service.

A.1.1. Preferences and Budget Constraint
The j th household maximizes j;t subject to P t C j;t + P I;t I j;t + B j;t RP;t = (R K;t u j;t a(u j;t )P I;t )K j;t 1 + W j;t N j;t + R t 1 B j;t 1 T j;t + A j;t Here, T j;t denotes lump-sum taxes net of pro…ts, P t denotes the price of consumption goods C j;t , P I;t denotes the price of investment goods I j;t , B j;t denotes one period risk-free bonds purchased in period t with gross return, R t . The object R K;t ; denotes the rental rate of capital services, K j;t 1 denotes the household's beginning of period t stock of capital, a(u j;t ) denotes the cost, in units of investment goods, of the capital utilization rate, u j;t and u j;t K j;t 1 denotes the household's period t supply of capital services. The functional form for the increasing and convex function, a ( ) ; is described below. All prices, taxes and pro…ts in the budget constraint are in nominal terms.
RP;t denotes the risk premium shock as used by Wouters (2003, 2007) which follows an AR(1) process.
The choice variables of the household are consumption, investment, risk-free bonds, capital utilization and the wage which is subject to Calvo wage setting frictions following EHL. The household understands that when choosing W j;t it must supply the amount of hours N j;t demanded. A j;t represents net proceeds of an asset that provides insurance against the idiosyncratic uncertainty associated with the Calvo wage-setting friction.
The parameters b; ! and control the degree of habit formation, the level of disutility from labor and the labor supply elasticity, respectively.
1= w h;t denotes an exogenous preference shifter for labor, scaled by 1= w . The scaling factor is the inverse slope of the linearized wage Phillips curve in terms of the marginal rate of substitution minus the real wage, i.e. 1/ w = 1= (1 w )(1 w ) w 1 1 (1+ w ) w . The scaling implies that h;t enters the linearized wage Phillips curve with a unit coe¢ cient. The same scaling is used in Smets and Wouters (2007) and in many other empirical papers. Note that h;t equals unity in steady state so that the scaling does not a¤ect the steady state. The scaling gets rid of the high negative correlation between the estimated standard deviations of wage markup shocks and the estimated parameter of the slope of the wage Phillips curve. Economically, h;t can be interpreted as a wage markup shock. We assume that h;t follows an ARMA(1,1) process.

A.1.2. Physical Capital
The representative household's stock of capital evolves as follows: The functional form for the increasing and convex adjustment cost function, S ( ) ; is described below.
t denotes a stationary investment-speci…c technology shock which follows an AR(1) process. We scale the investment-speci…c technology shock by the factor = k=i = 1= (1 (1 ) =( i )) such that the investment-speci…c technology shock enters the linearized law of motion for capital with a unit coe¢ cient. The same scaling was used in Adolfson, Laséen, Lindé and Villani (2007).

A.1.3. Wage Setting
Following EHL, we assume wage setting is subject to Calvo sticky wages. The wage setting problem is nearly identical to the one described in the stylized model, see section 2. 1 + ! subject to labor demand given by equation (7).

A.2. Firms and Price Setting
Final Goods Production. Like in the stylized model, the single …nal output good Y t is produced using a continuum of di¤erentiated intermediate goods Y f;t . Firms who produce the …nal output good are perfectly competitive in product and factor markets. The optimization problem is exactly the same as described in section 2.1.1. The resulting optimality conditions are (2), (3) and (4).
Intermediate Goods Production. Like in the stylized model, a continuum of intermediate goods and rents capital services K f;t and labor services N f;t for production. The form of the production function is Cobb-Douglas: Firms face perfectly competitive factor markets for renting capital and hiring labor. Thus, each …rm chooses K f;t and N f;t taking as given both the rental price of capital R K;t and the aggregate wage rate W t . Firms can costlessly adjust either factor of production. Thus, the standard static …rst-order conditions for cost minimization imply that all …rms have identical marginal cost per unit of output. Also, z t is unit-root neutral technology which we assume grows deterministically over time. t denotes a stationary neutral technology shock which we assume to follow an AR (1) process. t represents a …xed cost of production.
We assume that price setting is subject to Calvo sticky prices. The price setting problem is identical to the one described in section 2.1.1, except that we drop the discount factor shock in the medium-sized model. Speci…cally: where t+j is household marginal utility and demand Y f;t+j is given by equation (2). Similar to the canonical model, nominal marginal cost is given by: We add t to the model-consistent expression for marginal cost. It captures a variety of variations in marginal costs that are exogenous to the model. We assume that t follows an ARMA (1,1) process. Further, we scale t by the parameter 1= p : The scaling factor is the inverse slope of the linearized price Phillips curve in terms of real marginal cost, i.e. 1= p = 1= The scaling implies that t enters the linearized price Phillips curve with a unit coe¢ cient. The same scaling is used in Smets and Wouters (2007) and in many other empirical papers. Note that t equals unity in steady state so that the scaling does not a¤ect the steady state. The scaling gets rid of the high negative correlation between the estimated standard deviations of price markup shocks and the estimated parameter of the slope of the price Phillips curve.

A.3. Market Clearing and Functional Forms
Market clearing in the markets for labor and capital services require: Homogeneous output, Y t can be used to produce private or government consumption goods or investment goods. The production of the latter uses a linear technology in which one unit of Y t is transformed into t units of I t : Market clearing for …nal goods requires: R 1 0 C j;t dj and I t = R 1 0 I j;t dj: G t denotes government consumption. Perfect competition in the production of investment goods implies that the nominal price of investment goods equals the corresponding marginal cost: Aggregate output is given by: The variables p t and w t denote the price and wage dispersion terms de…ned in the main text.
The sources of growth in our model are unit-root neutral and investment-speci…c technological progress. Let: be the composite level to technology. In our model, Y t = t ; C t = t , (W t =P t ) = t and I t =( t t ) converge to constants in nonstochastic steady state. We also assume that ln z ln (z t =z t 1 ) and ln ln ( t = t 1 ), i.e. growth is deterministic in our model. Accordingly, let ln ln ( t = t 1 ) : In order to ensure a well-de…ned balanced growth path, we assume that the …xed cost of production follows t = t where is set such that pro…ts are zero along a balanced growth path.
We assume that the cost of adjusting investment takes the form: S (I t =I t 1 ) = 1 2 exp h p S 00 (I t =I t 1 ) i + exp h p S 00 (I t =I t 1 ) i 1: Here, and denote the unconditional growth rates of t and t . The value of I t =I t 1 in nonstochastic steady state is ( ): In addition, S 00 denotes the second derivative of S ( ), evaluated at steady state;and is a parameter to be estimated. It is straightforward to verify that S ( ) = S 0 ( ) = 0: We assume that the cost associated with setting capacity utilization is given by: where a and b are positive scalars. For a given value of a we select b so that the steady state value of u t is unity. The object a is a parameter to be estimated.

A.4. Fiscal and Monetary Policy
Government consumption G t is scaled by the level of composite unit-root technology t ; i.e. g t = G t = t : We assume that g t follows an exogenous AR(1) process. The government is assumed to balance its budget each period by adjusting lump-sum taxes. where 4 t = t t 1 t 2 t 3 denotes annual in ‡ation, Y t denotes real GDP de…ned as Y t = C t + I t = t + G t and Y f t denotes potential real GDP, i.e. real GDP if prices and wages are ‡exible. denotes the in ‡ation target of the central bank. r;t denotes the monetary policy shock. Finally, we impose the zero lower bound on the nominal interest rate by R t = max(1; R not t ):